---
title: "Transport, Wiedemann–Franz, and the Hall Effect"
module: Free-Electron Fermi Gas
moduleNumber: 5
lessonNumber: 3
order: 503
summary: >
  The relaxation-time picture displaces the Fermi sphere under an applied field
  and gives the electrical conductivity ne-squared-tau over m. The same electrons
  carry heat, and their ratio yields the Wiedemann–Franz law with the universal
  Lorenz number. A magnetic field bends the carriers into cyclotron orbits and
  produces the Hall voltage, whose sign reveals the charge of the carriers.
topics: [Free-Electron Fermi Gas]
draft: false
sources:
  - book: Ashcroft & Mermin
    ref: "Ch. 1 — The Drude Theory of Metals; Ch. 13 — The Semiclassical Theory of Conduction"
  - book: Kittel
    ref: "Ch. 6 — Free Electron Fermi Gas"
  - book: Tipler & Llewellyn
    ref: "Ch. 10 — Solid State Physics; §10-4 Quantum Theory of Conduction (Hall effect)"
---

The [Sommerfeld model](/condensed-matter/free-electron-fermi-gas/sommerfeld-model-and-heat-capacity)
gave the equilibrium electron gas: a filled Fermi sphere and a heat capacity set
by the thermal shell near $E_F$. Transport asks what happens when the gas is
pushed out of equilibrium by an electric field, a temperature gradient, or a
magnetic field. The same handful of electrons near the Fermi surface carry every
current, and one phenomenological parameter — the **relaxation time** $\tau$ —
ties the responses together. The payoff is a chain of results, culminating in the
Wiedemann–Franz law and the Hall effect, that made the free-electron picture
credible long before its microscopic justification.

## The relaxation-time approximation

Between collisions an electron of crystal momentum $\hbar\vec k$ obeys Newton's
law under the Lorentz force,

$$
\hbar\frac{\d\vec k}{\d t} = -e\left(\vec E + \vec v\times\vec B\right).
$$

Collisions — with lattice vibrations, impurities, other electrons — randomize
$\vec k$ on a timescale $\tau$. The **relaxation-time approximation** models this
by assuming the distribution $f(\vec k)$ relaxes back toward equilibrium
$f_0(\vec k)$ at a rate $1/\tau$:

$$
\frac{\partial f}{\partial t} = -\frac{f - f_0}{\tau} + \left(\frac{\partial f}{\partial t}\right)_{\text{drift}}.
$$

This is the Boltzmann transport equation with its collision integral replaced by a
single relaxation rate. In steady state the drift and relaxation terms balance.
The physical content is simplest in $k$-space: the field displaces the entire
Fermi sphere by a small amount $\delta\vec k = -e\vec E\tau/\hbar$, and collisions
try to push it back. The net drift velocity is $\vec v_d = \hbar\,\delta\vec
k/m = -e\vec E\tau/m$.

$$
% caption: An electric field along minus x displaces the whole Fermi sphere by a
% small delta-k opposite to the field. Electrons on the leading face have no
% occupied state ahead of them; scattering returns them to the trailing face.
% The net shift, exaggerated here, is what carries the current.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.0,0) -- (3.0,0) node[below right] {$k_x$};
  \draw[->, black] (0,-2.6) -- (0,2.6) node[left] {$k_y$};
  % equilibrium sphere centred at origin (dashed)
  \draw[black, thick, dashed] (0,0) circle (1.9);
  % displaced sphere shifted right
  \fill[acc!12] (0.55,0) circle (1.9);
  \draw[acc, very thick] (0.55,0) circle (1.9);
  \fill[black] (0.55,0) circle (1.6pt);
  \draw[black, ->, thick] (0,0) -- (0.55,0);
  \node[black, anchor=south, font=\scriptsize] at (0.28,0.05) {shift};
  % field arrow
  \draw[acc, ->, very thick] (-2.6,-2.05) -- (-1.3,-2.05) node[right, black] {$E$};
\end{tikzpicture}
$$

## Electrical conductivity

With $n$ electrons per unit volume the current density is $\vec j = -ne\vec v_d$,
and substituting $\vec v_d = -e\vec E\tau/m$ gives Ohm's law $\vec j = \sigma\vec
E$ with

$$
\sigma = \frac{ne^2\tau}{m}, \qquad \rho = \frac{1}{\sigma} = \frac{m}{ne^2\tau}.
$$

The form is identical to Drude's, but the interpretation is Sommerfeld's: the
electrons that shift are those at the Fermi surface, moving at $v_F$, so the mean
free path is $\ell = v_F\tau$ — set by the fast Fermi velocity, not the slow drift
or the classical thermal speed. Because only a perfect lattice fails to scatter,
$\tau$ is limited by deviations from periodicity. At high temperature phonon
scattering gives $\tau^{-1}\propto T$ and $\rho\propto T$; as $T\to 0$ the
residual impurity scattering leaves a temperature-independent floor, so
$\rho = \rho_{\text{phonon}}(T) + \rho_0$ — **Matthiessen's rule**, with the two
rates adding because the scattering probabilities add.

## Thermal conductivity and Wiedemann–Franz

The electron gas conducts heat as well as charge. Kinetic theory gives the
thermal conductivity of a gas of carriers as

$$
\kappa = \frac{1}{3}c_v\,v_F^2\,\tau = \frac{1}{3}c_v\,v_F\,\ell,
$$

where $c_v$ is the electronic heat capacity per unit volume. Using the Sommerfeld
result $c_v = \tfrac{\pi^2}{2}n k_B (T/T_F)$ together with $\tfrac12 m v_F^2 = E_F
= \tfrac32 k_B T_F$, the Fermi velocity and temperature cancel in a remarkable
way. Forming the ratio of thermal to electrical conductivity,

$$
\frac{\kappa}{\sigma} = \frac{\tfrac13 c_v v_F^2\tau}{ne^2\tau/m}
= \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2 T,
$$

so that $\kappa/\sigma T$ is a **universal constant** independent of the metal,
the carrier density, and the relaxation time:

$$
L \equiv \frac{\kappa}{\sigma T} = \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2
= 2.44\times10^{-8}\ \text{W}\,\Omega\,\text{K}^{-2}.
$$

> **Theorem (Wiedemann–Franz law).** For a degenerate free-electron metal in
> which charge and heat are carried by the same electrons with the same relaxation
> time, the ratio of thermal to electrical conductivity is $\kappa/\sigma = LT$
> with the **Lorenz number** $L = \tfrac{\pi^2}{3}(k_B/e)^2$, independent of
> material. The cancellation works because $\tau$ appears identically in both
> conductivities and because $c_v v_F^2$ is fixed by $E_F$.

The law holds impressively well for good metals near and above room temperature,
where scattering is elastic. It measures the Lorenz number to within a few percent
for copper, silver, gold, and the alkalis, and its main deviations — at
intermediate temperatures where phonon scattering of the heat current is
inelastic — are themselves diagnostic.

$$
% caption: The Lorenz number kappa over sigma T measured for several metals near
% room temperature, scattered narrowly about the free-electron value L0 (dashed).
% The near-constancy across metals with very different densities and relaxation
% times is the content of the Wiedemann–Franz law.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.2,0) node[below right] {metal};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {$L$};
  % L0 reference line
  \draw[black, dashed] (0,2.4) -- (6.8,2.4);
  \node[black, anchor=west, font=\scriptsize] at (5.6,2.62) {$L_0$};
  % bars near L0
  \foreach \x/\h/\name in {0.8/2.35/Cu, 1.9/2.28/Ag, 3.0/2.55/Au, 4.1/2.45/Pb, 5.2/2.15/Zn, 6.3/2.5/Cd}{
    \fill[acc!16] (\x-0.28,0) rectangle (\x+0.28,\h);
    \draw[acc, thick] (\x-0.28,0) rectangle (\x+0.28,\h);
    \node[black, anchor=north, font=\scriptsize] at (\x,-0.05) {\name};
  }
\end{tikzpicture}
$$

## Cyclotron motion and the Hall effect

Add a magnetic field $\vec B = B\hat z$ and no electric field. The Lorentz force
$-e\vec v\times\vec B$ bends the electron into a circle in the plane perpendicular
to $\vec B$, orbiting at the **cyclotron frequency**

$$
\omega_c = \frac{eB}{m},
$$

independent of the electron's speed or orbit radius. For $B = 1\ \text{T}$ this is
$1.76\times10^{11}\ \text{rad/s}$, or $28\ \text{GHz}$. Only if the electron can
complete an orbit between collisions — $\omega_c\tau \gg 1$, a clean sample in a
strong field — do the orbits matter for transport; otherwise scattering
interrupts them.

$$
% caption: In a magnetic field out of the page, the Lorentz force curves an
% electron into a cyclotron orbit of angular frequency omega-c equal to eB over m.
% The field points out of the page; the velocity and force are perpendicular.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % field out of page marks
  \foreach \p in {(-2.3,1.6),(-2.3,-1.6),(2.3,1.6),(2.3,-1.6)}{
    \draw[black] \p circle (3.2pt);
    \fill[black] \p circle (0.7pt);
  }
  \node[black, anchor=south, font=\scriptsize] at (-2.3,1.9) {$B$ out};
  % orbit
  \draw[acc, very thick] (0,0) circle (1.5);
  % electron and velocity
  \fill[black] (1.5,0) circle (2.2pt);
  \draw[acc, ->, thick] (1.5,0) -- (1.5,0.9) node[right, black] {$v$};
  \draw[black, ->, thick] (1.5,0) -- (0.75,0) node[above, font=\scriptsize] {force};
  \draw[black, ->] (0,0) -- (-1.06,1.06) node[midway, above, font=\scriptsize] {$r_c$};
\end{tikzpicture}
$$

Now the classic **Hall geometry**: current $I$ flows along $\hat x$ through a flat
bar, with $\vec B$ along $\hat z$. The magnetic force $-e\vec v\times\vec B$
pushes the drifting electrons toward one edge ($-\hat y$), charge piles up there,
and a transverse **Hall field** $E_y$ grows until it cancels the magnetic force in
steady state:

$$
-eE_y - e(v_x B) = 0 \;\Rightarrow\; E_y = -v_x B = \frac{j_x B}{ne}.
$$

The **Hall coefficient** is defined by $E_y = R_H\,j_x B$, giving

$$
R_H = \frac{E_y}{j_x B} = -\frac{1}{ne}.
$$

$$
% caption: Hall-bar geometry. Current I runs along x, field B along z. The
% magnetic force deflects carriers to one edge until the transverse Hall field
% Ey balances it. The sign of the measured Hall voltage fixes the sign of the
% charge carriers.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % bar
  \draw[black, thick] (0,0) rectangle (6.2,2.4);
  % current in
  \draw[acc, ->, very thick] (-1.0,1.2) -- (0,1.2) node[above left, black] {$I$};
  \draw[acc, ->, very thick] (6.2,1.2) -- (7.0,1.2);
  \node[acc, anchor=south, font=\scriptsize] at (3.1,1.25) {$j_x$};
  % B field out of page
  \draw[black] (3.1,0.55) circle (3.2pt);
  \fill[black] (3.1,0.55) circle (0.7pt);
  \node[black, anchor=west, font=\scriptsize] at (3.35,0.55) {$B$ out of page};
  % accumulated charge edges
  \node[black, anchor=south, font=\scriptsize] at (3.1,2.42) {negative edge};
  \node[black, anchor=north, font=\scriptsize] at (3.1,-0.02) {positive edge};
  % transverse field
  \draw[black, ->, thick] (5.4,2.1) -- (5.4,0.3) node[right, font=\scriptsize] {$E_y$};
  % Hall voltage leads
  \draw[black] (0.6,2.4) -- (0.6,3.0) node[above, font=\scriptsize] {$V_H$};
  \draw[black] (0.6,0) -- (0.6,-0.6);
\end{tikzpicture}
$$

The Hall coefficient measures the carrier density directly, and — its sharpest use —
the carrier **sign**. For free electrons $R_H < 0$, and for the alkali metals
the measured value matches $-1/ne$ within a few percent. But a number of
metals — beryllium, zinc, cadmium, aluminium — show a _positive_ Hall coefficient,
as if the current were carried by positive charges. Free-electron theory has no
explanation; the resolution is that in these metals the Fermi surface reaches the
Brillouin-zone boundary and the carriers behave as **holes** — a signal that the
periodic lattice cannot be ignored, taken up in
[band theory](/condensed-matter/band-theory/fermi-surfaces-and-semiclassical-dynamics).

## Magnetoresistance and the limits of the model

Solving the steady-state equations of motion with both $\vec E$ and $\vec B$ gives
the full **conductivity tensor**. For a single species of carrier the
free-electron model predicts **no magnetoresistance**: the diagonal resistivity
$\rho_{xx}$ is independent of $B$, because the Hall field exactly compensates the
transverse deflection. Real metals show a definite magnetoresistance, growing with
$B$ and often failing to saturate — another discrepancy that band structure, with
multiple carrier types and non-spherical Fermi surfaces, resolves.

The tally so far is favourable: the free-electron gas gives Ohm's law, the
Wiedemann–Franz law with the right Lorenz number, the cyclotron frequency, and the
Hall coefficient of the simple metals. Its failures — the positive Hall
coefficients, the magnetoresistance, the material-dependent $\gamma$ — all point at
the same missing ingredient, the periodic potential. Before that, one more purely
electronic effect deserves attention: how the gas **screens** a foreign charge and
oscillates collectively, the subject of the
[next lesson](/condensed-matter/free-electron-fermi-gas/screening-and-plasmons).